\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx \]
Optimal antiderivative \[ -\frac {B \left (d x +c \right )^{2} \left (b -\frac {4 d \left (b x +a \right )}{d x +c}\right )^{2}}{4 \left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )^{2}}-\frac {B \,d^{2} \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{2 \left (-a d +b c \right )^{3} g^{3} i}+\frac {2 b d \left (d x +c \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )}-\frac {b^{2} \left (d x +c \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{2 \left (-a d +b c \right )^{3} g^{3} i \left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (\frac {b x +a}{d x +c}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{3} g^{3} i} \]
command
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (2 i \, B e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 i \, A e^{3} + i \, B e^{3}\right )} {\left (d x + c\right )}^{2} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2}}{4 \, {\left (b x e + a e\right )}^{2} g^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________